Solving -1.2 X 0.8 ÷ (-0.03) A Step-by-Step Guide

In the realm of mathematics, precision and order of operations are paramount. When faced with an expression like $-1.2 \times 0.8 \div(-0.03)$, it's essential to navigate the calculations systematically to arrive at the correct answer. This article delves into the step-by-step process of solving this mathematical expression, providing a clear and comprehensive understanding of the underlying principles. We will explore the rules of arithmetic, the significance of negative signs, and the importance of following the correct order of operations – often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). By carefully dissecting each step, we aim to not only provide the solution but also to illuminate the mathematical reasoning behind it.

Understanding the Order of Operations

Before we plunge into the calculations, it's crucial to understand the order of operations. This set of rules dictates the sequence in which mathematical operations should be performed to ensure consistency and accuracy in calculations. The acronym PEMDAS serves as a helpful mnemonic: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). In our expression, $-1.2 \times 0.8 \div(-0.03)$, we have multiplication and division. According to PEMDAS, these operations should be performed from left to right. This means we'll first multiply -1.2 by 0.8, and then divide the result by -0.03. Neglecting this order can lead to a completely different, and incorrect, answer. Therefore, a solid grasp of PEMDAS is fundamental to tackling any mathematical expression involving multiple operations. This foundational understanding not only helps in solving individual problems but also builds a stronger mathematical intuition, allowing for a more confident approach to complex calculations.

Step 1: Multiplication – $-1.2 \times 0.8$

The first step in solving the expression $-1.2 \times 0.8 \div(-0.03)$ is to perform the multiplication: $-1.2 \times 0.8$. When multiplying decimals, it's often helpful to initially ignore the decimal points and perform the multiplication as if they were whole numbers. So, we multiply 12 by 8, which equals 96. Now, we need to account for the decimal places. In -1.2, there is one decimal place, and in 0.8, there is also one decimal place. This means the product will have a total of two decimal places. Therefore, we place the decimal point two places from the right in 96, resulting in 0.96. However, we also need to consider the signs. A negative number multiplied by a positive number yields a negative result. Hence, $-1.2 \times 0.8 = -0.96$. This step showcases the importance of paying attention to both the numerical values and the signs involved in the calculation. A misstep here can cascade through the rest of the problem, leading to an incorrect final answer. Thus, precision in this initial multiplication is key.

Step 2: Division – $-0.96 \div(-0.03)$

Having completed the multiplication, the next step is to perform the division: $-0.96 \div(-0.03)$. Dividing decimals can sometimes seem daunting, but it can be simplified by eliminating the decimal points. To do this, we can multiply both the dividend (-0.96) and the divisor (-0.03) by 100. This shifts the decimal point two places to the right in both numbers, transforming the division problem into $-96 \div(-3)$. Now, the division is much simpler. We know that 96 divided by 3 is 32. The final consideration is the signs. A negative number divided by a negative number results in a positive number. Therefore, $-96 \div(-3) = 32$. This step highlights a crucial aspect of division: dealing with decimals and signs. By understanding how to manipulate decimals and apply the rules of signs, we can convert seemingly complex division problems into manageable calculations. The resulting positive value of 32 completes this division step, setting the stage for the final answer.

Final Result and Conclusion

After meticulously performing the multiplication and division in the expression $-1.2 \times 0.8 \div(-0.03)$, we have arrived at the final result: 32. This solution is the culmination of a series of steps, each requiring careful attention to detail and a firm understanding of mathematical principles. We began by emphasizing the importance of the order of operations (PEMDAS), ensuring that multiplication was performed before division. We then tackled the multiplication of -1.2 and 0.8, correctly handling the signs and decimal places to arrive at -0.96. The subsequent division of -0.96 by -0.03 involved converting the decimal division into a whole number division, yielding the positive result of 32. This entire process underscores the interconnectedness of mathematical operations and the necessity of a systematic approach. A single error in any step can significantly alter the final outcome. Therefore, practice and a keen eye for detail are essential for mastering these types of calculations. The final answer of 32 not only provides a numerical solution but also reinforces the importance of precision and methodical problem-solving in mathematics.

Understanding the Expression -1.2 x 0.8 ÷ (-0.03)

When confronted with a mathematical expression like -1.2 x 0.8 ÷ (-0.03), it’s crucial to dissect it methodically. This expression involves multiplication and division, operations that require a precise understanding of numerical relationships and order of operations. At first glance, the presence of decimals and negative signs might seem intimidating. However, by breaking down the problem into manageable steps and applying the correct mathematical principles, we can unravel the complexity and arrive at an accurate solution. The key lies in adhering to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that we perform the calculations in the correct sequence, avoiding errors and maintaining the integrity of the mathematical process. This foundational understanding is not just about solving this specific problem; it's about developing a broader mathematical fluency that enables us to tackle a wide range of numerical challenges with confidence and precision.

Importance of Order of Operations (PEMDAS/BODMAS)

The cornerstone of solving any mathematical expression involving multiple operations is the order of operations, commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This set of rules acts as a roadmap, guiding us through the steps in the correct sequence. In the context of -1.2 x 0.8 ÷ (-0.03), we have multiplication and division. According to PEMDAS/BODMAS, these operations have equal priority and should be performed from left to right. This means we first tackle the multiplication of -1.2 and 0.8, and then divide the result by -0.03. Ignoring this order can lead to a drastically different, and incorrect, answer. For instance, if we were to divide 0.8 by -0.03 first, we would end up with a completely different intermediate result, ultimately leading to a wrong final answer. Therefore, a firm grasp of PEMDAS/BODMAS is not just a recommendation; it's a necessity for accurate mathematical calculations. It provides a structured approach that minimizes errors and ensures consistency in problem-solving.

Step-by-Step Solution: Breaking Down the Calculation

To solve -1.2 x 0.8 ÷ (-0.03) accurately, let's break it down into manageable steps:

Step 1: Multiply -1.2 by 0.8

This initial step involves the multiplication of two decimal numbers, one of which is negative. To perform this multiplication, we can initially ignore the decimal points and multiply 12 by 8, which gives us 96. Now, we need to account for the decimal places. -1.2 has one decimal place, and 0.8 has one decimal place, making a total of two decimal places in the product. Therefore, we place the decimal point two places from the right in 96, resulting in 0.96. Finally, we consider the signs. A negative number multiplied by a positive number yields a negative result. So, -1.2 x 0.8 equals -0.96. This step highlights the importance of careful multiplication and correct handling of signs, which are crucial for an accurate result. A mistake here would propagate through the rest of the calculation, leading to an incorrect final answer.

Step 2: Divide -0.96 by -0.03

The second step is to divide -0.96 by -0.03. Dividing decimals can be simplified by eliminating the decimal points. We can achieve this by multiplying both the dividend (-0.96) and the divisor (-0.03) by 100. This effectively shifts the decimal point two places to the right in both numbers, transforming the division problem into -96 ÷ -3. Now, the division is straightforward. 96 divided by 3 is 32. The final consideration is the signs. A negative number divided by a negative number results in a positive number. Therefore, -96 ÷ -3 equals 32. This step demonstrates a useful technique for simplifying decimal division and reinforces the rules for dividing signed numbers. The resulting positive value of 32 completes the calculation, providing the final answer.

Final Answer and Conclusion

After meticulously following the order of operations and performing the calculations step by step, we arrive at the final answer for -1.2 x 0.8 ÷ (-0.03), which is 32. This result underscores the importance of a systematic approach in mathematics. By breaking down the problem into smaller, more manageable steps, we were able to navigate the complexities of decimals, negative signs, and the order of operations. The initial multiplication of -1.2 and 0.8 gave us -0.96, and the subsequent division of -0.96 by -0.03 resulted in the positive value of 32. This journey through the calculation highlights the interconnectedness of mathematical operations and the need for precision at each stage. A single error in any step could have led to a different, and incorrect, final answer. Therefore, practice, a clear understanding of mathematical principles, and attention to detail are essential for achieving accuracy in calculations. The final answer of 32 serves not just as a numerical solution but as a testament to the power of methodical problem-solving in mathematics.

Common Mistakes to Avoid When Solving This Type of Problem

When tackling mathematical expressions like -1.2 x 0.8 ÷ (-0.03), certain common mistakes can lead to incorrect answers. Being aware of these pitfalls can significantly improve accuracy and problem-solving skills. One frequent error is neglecting the order of operations (PEMDAS/BODMAS). Forgetting to perform multiplication and division from left to right can lead to incorrect intermediate results and a wrong final answer. Another common mistake is mishandling decimal points. Errors in placing the decimal point during multiplication or division can drastically alter the result. Similarly, mistakes in applying the rules of signs can be detrimental. For instance, incorrectly determining the sign of the product or quotient can lead to an incorrect answer. Another potential pitfall is rushing through the calculations without double-checking each step. Simple arithmetic errors can easily occur if calculations are not performed carefully and systematically. By recognizing these common mistakes and taking proactive steps to avoid them, we can significantly enhance our ability to solve mathematical expressions accurately and efficiently.

Practical Applications of This Type of Calculation

Understanding how to solve expressions like -1.2 x 0.8 ÷ (-0.03) extends beyond the realm of pure mathematics and has numerous practical applications in various fields. In finance, such calculations might be used to determine the return on an investment after accounting for losses and gains. For example, calculating percentage changes or compound interest often involves similar operations with decimals and negative numbers. In science and engineering, these types of calculations are essential for unit conversions, determining physical quantities, and modeling real-world phenomena. For instance, converting measurements between different units or calculating the force exerted on an object might require multiplying and dividing decimal values. Even in everyday situations, these skills come in handy. When calculating discounts, splitting bills, or figuring out proportions, the ability to accurately perform multiplication and division with decimals is invaluable. Therefore, mastering these mathematical concepts is not just an academic exercise; it's a practical skill that empowers us to solve a wide range of problems in diverse contexts.

Practice Problems to Sharpen Your Skills

To solidify your understanding of how to solve expressions like -1.2 x 0.8 ÷ (-0.03), engaging in practice problems is essential. The more you practice, the more confident and proficient you'll become in applying the correct mathematical principles and avoiding common mistakes. Try solving similar expressions with varying numbers and operations. For example, you could tackle problems like -2.5 x 1.5 ÷ (-0.05) or 0.75 x (-0.4) ÷ 0.02. Varying the complexity of the problems, such as including more operations or parentheses, can further challenge your skills and deepen your understanding. Additionally, it's beneficial to work through problems with and without the aid of a calculator. This helps you not only verify your answers but also develop your mental math abilities. Remember to focus on each step of the calculation, paying close attention to the order of operations, decimal points, and signs. By consistently practicing and analyzing your solutions, you can sharpen your mathematical skills and build a strong foundation for tackling more complex problems in the future.

Breaking Down the Mathematical Expression -1.2 x 0.8 ÷ (-0.03)

Mathematical expressions can often appear daunting at first glance, but with a systematic approach, they can be easily解. The expression -1.2 x 0.8 ÷ (-0.03) is a prime example. It involves a combination of multiplication and division operations, along with decimal numbers and negative signs. To accurately solve this expression, we need to break it down into smaller, manageable steps, carefully applying the rules of arithmetic and the order of operations. The presence of decimals adds a layer of complexity, as does the presence of negative signs, which require careful attention to ensure the correct application of sign rules. However, with a clear methodology and a focus on detail, we can navigate these challenges and arrive at the correct solution. This process is not just about finding the answer; it's about developing a robust problem-solving strategy that can be applied to a wide range of mathematical challenges.

Applying the Order of Operations: A Step-by-Step Approach

The order of operations is the fundamental principle that governs how we solve mathematical expressions involving multiple operations. It ensures that we perform calculations in the correct sequence, leading to a consistent and accurate result. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) serves as a handy mnemonic for remembering this order. In the expression -1.2 x 0.8 ÷ (-0.03), we encounter multiplication and division. According to PEMDAS, these operations have equal priority and should be performed from left to right. This means our first step is to multiply -1.2 by 0.8, and then we will divide the result by -0.03. Strictly adhering to this order is crucial; deviating from it can lead to a completely different, and incorrect, answer. The order of operations is not merely a convention; it's a mathematical necessity that ensures consistency and avoids ambiguity in calculations. A thorough understanding of this principle is essential for anyone seeking to master mathematical problem-solving.

Detailed Solution: Multiplying and Dividing Decimals and Signed Numbers

Let's delve into the detailed steps for solving the expression -1.2 x 0.8 ÷ (-0.03):

Step 1: Multiply -1.2 by 0.8

To multiply -1.2 by 0.8, we can initially disregard the decimal points and multiply 12 by 8, which yields 96. Now, we need to account for the decimal places. -1.2 has one decimal place, and 0.8 has one decimal place, making a total of two decimal places in the product. Therefore, we place the decimal point two places from the right in 96, resulting in 0.96. The final consideration is the signs. A negative number multiplied by a positive number results in a negative number. So, -1.2 x 0.8 equals -0.96. This step underscores the importance of careful decimal placement and the correct application of sign rules. A mistake in either of these areas would lead to an inaccurate intermediate result and ultimately a wrong final answer.

Step 2: Divide -0.96 by -0.03

The second step involves dividing -0.96 by -0.03. To simplify this division, we can eliminate the decimal points by multiplying both the dividend (-0.96) and the divisor (-0.03) by 100. This transforms the division problem into -96 ÷ -3. Now, the division is straightforward. 96 divided by 3 is 32. We must also consider the signs. A negative number divided by a negative number results in a positive number. Therefore, -96 ÷ -3 equals 32. This step illustrates a useful technique for simplifying decimal division and reinforces the rules for dividing signed numbers. The resulting positive value of 32 completes the calculation, providing the final answer.

Conclusion: Arriving at the Final Answer and Understanding the Process

Having meticulously followed the order of operations and performed the calculations step by step, we arrive at the final answer for the expression -1.2 x 0.8 ÷ (-0.03), which is 32. This result is not just a numerical value; it's a testament to the power of systematic problem-solving in mathematics. By breaking down the problem into smaller, more manageable steps, we were able to navigate the complexities of decimals, negative signs, and the order of operations. The initial multiplication of -1.2 and 0.8 gave us -0.96, and the subsequent division of -0.96 by -0.03 resulted in the positive value of 32. This journey through the calculation highlights the interconnectedness of mathematical operations and the need for precision at each stage. A single error in any step could have led to a different, and incorrect, final answer. Therefore, practice, a clear understanding of mathematical principles, and attention to detail are essential for achieving accuracy in calculations. The final answer of 32 serves not just as a solution to a specific problem but as an affirmation of the importance of methodical thinking in mathematics and beyond.